Bounds on the localization number

Abstract

We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph G is called the localization number and is written ζ(G). We settle a conjecture of [5] by providing an upper bound on the localization number as a function of the chromatic number. In particular, we show that every graph with ζ(G) ≤ k has degeneracy less than 3 k and, consequently, satisfies χ(G) ≤ 3 ζ(G) . We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.

The localization game on Cartesian products

“…By Corollary 2.12, ζ(C n ) ≥ log 2 2 = 1 and by Theorem 3.13, ζ(C n ) = 1. Therefore even cycles of order at least eight prove the tightness of Corollary 2.12 and so providing an affirmative answer to the question in [3] on whether the bound is tight.…”

“…The technique in the above proof is based on the technique used by Bonato et al [3] in their proof of Proposition 2.10.…”

The localization number of designs

“…See, for example, [2]. The localization game was first introduced for one cop by Seager [24,25] and was further studied in [6,13,14]. Interestingly, the localization number is related to the metric dimension of a graph, in a way that is analogous to how the cop number is bounded above by the domination number.…”

“…In [17], the localization number was studied for binomial random graphs with diameter 2, with further work done in this direction was done in [16,17]. Bonato and Kinnersley [6] studied the localization number of graphs based on their degeneracy. In [6], they resolved a conjecture of [11] relating ζ G ( ) and the chromatic number; further, they proved that the localization number of outerplanar graphs is at most 2, and they proved an asymptotically tight upper bound on the localization number of the hypercube.…”

“…The degeneracy of a graph G is the maximum, over all subgraphs H of G, of the minimum degree of H. There is an immediate lower bound for BIBD's with general λ, based on degeneracy results from [6].…”